In the shaping, cutting or machining techniques with which the invention is concerned, a wire-type or band-type elongated tool has been hitherto employed which is composed of a high-tensile-strength material which may have abrasive particles or grit pre-bonded thereon by electrodeposition or other bonding process so as to serve as cutting or grinding edges on the periphery of the carrier wire or band against a workpiece which is brought into contact with and urged against the axially moving wire or band tool. Alternatively, abrasive grits may be used in suspension with a machining fluid which is introduced into a machining region comprised of an axially moving wire or band tool and a workpiece urged thereagainst to undergo mechanical abrading actions by the suspended abrasive grit. It has also been proposed to use an electrically conductive wire or band as an axially travelling electrode in electrical-discharge or electrolytic machining processes with the machining fluid being constituted by a liquid dielectric or a liquid electrolyte. These latter processes may be carried out individually or in combination through the use of a suitable machining fluid and, also in combination with the use of abrasives prebonded to the electrode or in the form of a suspension as described. As the cutting proceeds, the workpiece is displaced relative to the moving wire or band tool along a prescribed path to impart to the workpiece a desired cut or shape.